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Definition 2 (Simple Hyper model). Asimple hyper model is a tuple ( S,
,
,
,V ) where:
- ( S,
,V ) is a Kripke model w.r.t. ʣ .
2 S is a labelled binary relation fromastate to a setofstates.
-
S
×
ʣ
×
2 S is a labelled binary relation fromastate to a setofstates.
-
S
×
ʣ
×
S : s a
T such that s a
- for all s
S, T
T implies thatthere exists t
t .
S : s a
S : s a
- for all s
S, T
T implies thatforall t
t implies t
T .
In (simple) hyper models,
represents the actual transitions between the states, and
and
represent the available imperfect procedural information to an agent. The
last two conditions are crucial to guarantee the correctness of procedural information
in the model. Note that this model is from the modeller's point of view, and the agent's
knowledge only depends on
, as it will become clear in the se-
mantics of the logic.Hereweinclude the actual transitions in order to validate whether
ourlogic, to be defined later, is a proper epistemic logic, e.g., whether everything the
agent knows is actually true. When representing the agent's procedural information
only, we can simply leave out the actual transitions given
and
but not
and
are reliable.
a
Note that s
denotes 'negative' information: there is no a -transition from s .On
the other hand, it is impossible to have s a
due to the first correctness condition.
. As an example, recall
the model we mentioned at the beginning of this section (now with the actual transi-
tions):
Note that, the transitions
and
are not defined by
where:
A
B
b
- S =
{
A, B, C, D
}
,
b
b
-
=
{
( A, b, B ) , ( A, t, D ) , ( C,t,D )
}
,
t
-
=
{
( A, b,
{
B,C
}
) , ( C,t,
{
D
}
)
}
,
-
=
{
( B,b,
{
C
}
)
}
,
t
- for all s, v
∈{
A, B, C, D
}
, p s
V ( v ) iff s = v .
D
C
t
It is easy to verify that the last two correctness conditions are satisfied, e.g., for
b
{B,C}
b
→ B . On the other hand, although A
t
→ D , there is no
t
A
we have A
nor t
from A to D .
Remark 1. Some readers may wonder whether further conditions on
should
apply, to which we will come back in Section 4. For now, let us keep everything simple
to understand the merit of the framework.
and
A fragment of EPDL is usedtotalkabout the knowledgeofasingle agent on simple
hyper models:
Definition 3 (Epistemic Action Language EAL). Givenacountable set of proposi-
tional variables P , a finite sets of atomic actions ʣ ,theformulas of EAL are given
by:
ˆ ::=
|
p
ˆ
|
( ˆ
ˆ )
|
|
a
ˆ
where p
P and a
ʣ .
 
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